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- W4383173722 abstract "Let $d$ be a positive integer. For a finite set $X subseteq mathbb{R}^d$, we define its integer cone as the set $mathsf{IntCone}(X) := { sum_{x in X} lambda_x cdot x mid lambda_x in mathbb{Z}_{geq 0} } subseteq mathbb{R}^d$. Goemans and Rothvoss showed that, given two polytopes $mathcal{P}, mathcal{Q} subseteq mathbb{R}^d$ with $mathcal{P}$ being bounded, one can decide whether $mathsf{IntCone}(mathcal{P} cap mathbb{Z}^d)$ intersects $mathcal{Q}$ in time $mathsf{enc}(mathcal{P})^{2^{mathcal{O}(d)}} cdot mathsf{enc}(mathcal{Q})^{mathcal{O}(1)}$ [J. ACM 2020], where $mathsf{enc}(cdot)$ denotes the number of bits required to encode a polytope through a system of linear inequalities. This result is the cornerstone of their XP algorithm for BIN PACKING parameterized by the number of different item sizes. We complement their result by providing a conditional lower bound. In particular, we prove that, unless the ETH fails, there is no algorithm which, given a bounded polytope $mathcal{P} subseteq mathbb{R}^d$ and a point $q in mathbb{Z}^d$, decides whether $q in mathsf{IntCone}(mathcal{P} cap mathbb{Z}^d)$ in time $mathsf{enc}(mathcal{P}, q)^{2^{o(d)}}$. Note that this does not rule out the existence of a fixed-parameter tractable algorithm for the problem, but shows that dependence of the running time on the parameter $d$ must be at least doubly-exponential." @default.
- W4383173722 created "2023-07-05" @default.
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- W4383173722 date "2023-07-01" @default.
- W4383173722 modified "2023-09-27" @default.
- W4383173722 title "Detecting Points in Integer Cones of Polytopes is Double-Exponentially Hard" @default.
- W4383173722 doi "https://doi.org/10.48550/arxiv.2307.00406" @default.
- W4383173722 hasPublicationYear "2023" @default.
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